Banach's Fixed Point Theorem

Banach's Fixed Point Theorem states that in a complete space with a contraction mapping (a function that brings points closer together), there is exactly one point that remains unchanged when the function is applied. This fixed point can be found by starting from any initial point and repeatedly applying the function; as you do, the sequence will get closer to this unique point. The theorem guarantees both the existence and uniqueness of the fixed point, and that an iterative process will reliably find it. It is fundamental in fields like analysis and computer science for solving equations and ensuring convergence.